Lattice gas model in random medium and open boundaries: hydrodynamic and relaxation to the steady state

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions $d \ge 3$, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a non linear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation

A particle system with explosions: law of large numbers for the density of particles and the blow-up time

Consider a system of independent random walks in the discrete torus with creation-annihilation of particles and possible explosion of the total number of particles in finite time. Rescaling space and rates for diffusion/creation/annihilation of particles, we obtain a stong law of large numbers for the density of particles in the supremum norm. The limiting object is a classical solution to the semilinear heat equation u_t =u_{xx} + f(u). If f(u)=u^p, 1<p \le 3, we also obtain a law of large numbers for the explosion time

Phase segregation dynamics for the Blume-Capel model with Kac interaction

We consider the Glauber and Kawasaki dynamics for the Blume-Capel spin model with weak long-range interaction on the infinite lattice: a ferromagnetic d-dimensional lattice system with the spin variable sigma taking values in {-1, 0, 1} and pair Kac potential gamma(d)(gamma(\ i - j \)), gamma > 0, i,j is an element of Z(d). The Kawasaki dynamics conserves the empirical averages of sigma and sigma(2) corresponding to local magnetization and local concentration. We study the behaviour of the system under the Kawasaki dynamics on the spatial scale gamma(-1) and time scale gamma(-2). We prove that the empirical averages converge in the limit gamma --> 0 to the solutions of two coupled equations, which are in the form of the flux gradient for the energy functional. In the case of the Glauber dynamics we still scale the space as gamma(-1) but look at finite time and prove in the limit of vanishing gamma the law of large number for the empirical fields. The limiting fields are solutions of two coupled nonlocal equations. Finally, we consider a nongradient dynamics which conserves only the magnetization and get a hydrodynamic equation for it in the diffusive limit which is again in the form of the flux gradient for a suitable energy functional. (C) 2000 Elsevier Science B.V. All rights reserved

Phase segregation dynamics for the Blume-Capel model with Kac interaction

We consider the Glauber and Kawasaki dynamics for the Blume-Capel spin model with weak long-range interaction on the infinite lattice: a ferromagnetic d-dimensional lattice system with the spin variable [sigma] taking values in {-1,0,1} and pair Kac potential . The Kawasaki dynamics conserves the empirical averages of [sigma] and [sigma]2 corresponding to local magnetization and local concentration. We study the behaviour of the system under the Kawasaki dynamics on the spatial scale [gamma]-1 and time scale [gamma]-2. We prove that the empirical averages converge in the limit [gamma]-->0 to the solutions of two coupled equations, which are in the form of the flux gradient for the energy functional. In the case of the Glauber dynamics we still scale the space as [gamma]-1 but look at finite time and prove in the limit of vanishing [gamma] the law of large number for the empirical fields. The limiting fields are solutions of two coupled nonlocal equations. Finally, we consider a nongradient dynamics which conserves only the magnetization and get a hydrodynamic equation for it in the diffusive limit which is again in the form of the flux gradient for a suitable energy functional.Interacting particle and spin systems Kac potential Hydrodynamic limits Phase segregation

Large deviations from the macroscopic equation for a particle systems with external random field and Kac type interaction

We consider a lattice gas in a periodic $d-$ dimensional lattice of width \g^{-1}, \g>0, interacting via a Kac's type interaction, with range \frac 1\g and strength \g^d, and under the influence of a random potential given by independent, bounded, random variables with translational invariant distribution. The system evolves through a conservative dynamics, i.e. particles jump to nearest neighbor empty sites, with rates satisfying detailed balance with respect to the equilibrium measures. In [21] it has been shown that rescaling space as \g^{-1} and time as \g^{-2}, in the limit \g \downarrow 0, for dimensions $d\ge 3$, the macroscopic density profile \r satisfies, a.s. with respect to the random field, a nonlinear integral partial differential equation, having the diffusion matrix determined by the statistical properties of the external random field. Here we show an almost sure (with respect to the random field) large deviations principle for the empirical measures of such a process. The rate function, which depends on the statistical properties of the external random field, is lower semicontinuous and has compact level sets

Boundary driven Kawasaki process with long range interaction: dynamical large deviations and steady states

A particle system with a single locally-conserved field (density) in a bounded interval with different densities maintained at the two endpoints of the interval is under study here. The particles interact in the bulk through a long-range potential parametrized by β 0 and evolve according to an exclusion rule. It is shown that the empirical particle density under the diffusive scaling solves a quasilinear integro-differential evolution equation with Dirichlet boundary conditions. The associated dynamical large deviation principle is proved. Furthermore, when β is small enough, it is also demonstrated that the empirical particle density obeys a law of large numbers with respect to the stationary measures (hydrostatic). The macroscopic particle density solves a non-local, stationary, transport equation

Hydrostatics and dynamical large deviations of boundary driven gradient symmetric exclusion processes

We prove the hydrostatics of boundary driven gradient exclusion processes, Fick's law and we present a simple proof of the dynamical large deviations principle which holds in any dimension.Boundary driven exclusion processes Stationary nonequilibrium states Hydrostatics Fick's law Large deviations